In the paper we discuss the following type congruences: $$ \biggl ({np^k\atop mp^k}\biggr ) \equiv \left (m \atop n\right ) \pmod {p^r}, $$ where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren's and Jacobsthal's type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.
A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \pmod {p^4}$. For such a prime $p$, we establish an expression for ${2p-1\choose p-1}\pmod {p^8}$ given in terms of the sums $R_i:=\sum _{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$ \left ({2p-1\atop p-1}\right ) \equiv 1 -2p \sum _{k=1}^{p-1}\frac {1}{k} -2p^2\sum _{k=1}^{p-1}\frac {1}{k^2}\pmod {p^7}. $$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique of Helou and Terjanian, the above congruence is given as an expression involving the Bernoulli numbers.